The goal of the proposed research is to develop tools for the identification and characterization of genes underlying traits with complex patterns of inheritance. This knowledge is fundamental to the understanding, of diagnosis, treatment and prevention and of many common disease. Basic population-genetic questions, regarding the origin and maintenance of variations in complex traits, are also fundamentally connected to the characterization of the underlying genes. Among the various approaches to mapping complex-trait genes, association mapping is perhaps the most promising, yet least well understood. Optimal association to mapping complex-trait genes, association mapping is perhaps the most promising, yet least well understood. Optimal association-mapping methods for complex-trait gene detection and localization must be based on explicit population genetic models. Three interacting research projects are designed to meet the critical theoretical and practical hurdles. First, likelihood methods, based on population genetic models for patterns of haplotype structure in population samples will be developed; likelihood- based data analysis will be carried out using analytical, numerical and simulation methods. In principle, any number of markers (and complex- trait genes) can be considered, but computationally accessible solutions may only be feasible for two, three and four loci. At this level of resolution, the most significant questions concerning genomic localization and allelic effects can be addressed. Many populations have a history or recent admixture, which is known to creature spurious associations between explicitly utilize admixture information in mapping and characterization of effects. The increasing numbers of efficiently scorable markers (most of which are unlinked) are a potential source for accurate detection of admixture effects. In the second main project, principle components decompositions of the linkage. Applications of corrections of admixture effects in an association mapping context will be developed and evaluated. In the third main project, population-genetics simulation programs (based on coalescent processes and incorporating diploidy, admixture and complex-trait genes) will be developed to evaluate the tools for association mapping. Access to such simulations and to the programs themselves will be established on the world-wide- web.